In the XY-Plane does line l intersect the graph of y=x^2?

(1) Line l passes through the point (4, -8). Consider the diagram below:As you can see line passing through (4, -8) may or may not intersect with the graph of y=x^2. Not sufficient.

(2) Line l passes through the point (-4, 16). Since (-4)^2=16, then point (-4, 16) is ON the graph of y=x^2, thus line passing through this point intersects the graph of y=x^2. Sufficient.

Answer: B.

Hope it's clear.
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1) This is clearly insufficient. It is possible that the line might intersect the curve at 4,16 ( if it has slope 0, like x=4), or might not intersect at all, like line y=-8.

2) This is sufficient as the line satisfies the equation of the curve y=x^2, 16=(-4)^2, therefore the line and curve intersect at this point.

Answer should be B.

Hi Bunuel/Karishma,

I was able to solve this question by passing values in the equation y=x^2 and have found the correct answer. For reviewing the question, I googled it and ve found that Gmat instructors are rating this question -HARD.
Could you please give me an idea what makes this question Hard. I'm bit skeptical about my approach now..

Thanks
H

Personally I wouldn't rate this question as hard. I think its difficulty level is ~600, not more.
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GMAC rates this Q as hard , if you ask me there is a reason for it.

There is a line passing through (-4, 16) that will be tangent to the curve at (-4, 16). Can you say that a tangent intersects a curve. The literature says the tangent touches a curve, not sure if touch and intersect are the same thing. Intersect means dividing in sects/sections. However the tangent lies totally outside the curve.

This fact would make B insufficient as there is one line the tangent that does not intersect the curve. Hence the answer would be C and not B.

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If however you take the mathematical definition that an equation of line and curve can be solved for one or more points then the line intersects the curve, then B will be sufficient. That's how i was able to digest the solution .
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It's definitely rated as a hard question.

Touching and intersecting mean the same thing . This is because the lines share a common point.

Hence, you arrive at B

Thanks,
Ankit

\(y=x^2\) is a parabola with vertex at (0,0) and upward......

(a) Line passes through (4,-8); if line is parallel to X-axis it can never intersect the parabola.... if line is parallel to Y-axis it will... Not Sufficient
(b) Line passes through (-4,16); observe that this point is on \(y=x^2\) which means line is intersecting parabola at the point.... It may or maynot intersect \(y=x^2\) at some other point, we don't bother about it.... because question asks for if the line is intersecting \(y=x^2\) or not... so just a yes or no question.... in this case it is intersecting the graph... so sufficient....

Answer is B

Hi Gurus,

My question may be absurd but please help me understand the concept here. When I see y=x^2 i do not see it as an Upright parabola but a parabola drawn towards positive x axis . Say y=-3 then x^2 = 9 so (-3,9) is on the parabola, similarly (-2,4),(-1,1),(0,0),(1,1) and (2.4) all should form the parabola with function y=X^2.

Now if line l passes through (-4,16)- statement B, i see it as x co-ordinate as -4 and y co-ordinate as 16 then this point does not lie on the parabola. Can anyone please explain where am I at fault?

Thanks in advance.

Arun

I think you should brush-up fundamental on coordinate geometry:


As for your question, check the graph of y=x^2 and the point (-4, 16) on it:To check whether (-4, 16) is on y=x^2, substitute x=-4 there and check whether y comes out to be 16: (-4)^2=16, so (-4, 16) is on y=x^2.

Hope it helps.
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Moreover, IMO what makes this question hard is the fact that most believe that St1 is clearly insufficient and I dont think that's the case. Here we need to ensure that point (4, -8) does not lie inside the parabola. If the point lies inside the parabola it will cross the parabola no matter what. Thus this verification will involve 2 steps drawing the parabola and locating the point. This will take up some time.
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I understand the everything concerning how line l intersects y=x^2 at (-4,16) and not (4,-8), but I am confused as to what the question is asking. As a data sufficiency problem, are we not simply supposed to select the statements that allow us to reach an answer? For example, this question specifically asks whether line intersects y=x^2; then, is the first statement not sufficient NOT simply because (4,-8) isn't on y=x^2 but because even if it's not, we don't know if it intersects at another point (i.e. a vertical line at x=4 vs. a horizontal line at y=-8)? Just trying to make sure I'm not getting the right answer for the wrong reason.

Thanks in advance!

There are two kinds of data sufficient questions: YES/NO DS questions and DS questions which ask to find a value.

In Yes/No Data Sufficiency questions, statement is sufficient if the answer is “always yes” or “always no” while a statement is insufficient if the answer is "sometimes yes" and "sometimes no".

When a DS question asks about the value of some variable, then the statement is sufficient ONLY if you can get the single numerical value of this variable.


The original question is an Yes/No Data Sufficiency question. And the first statement is not sufficient because we can have both yes and no answers to the question. See my post with diagram in it.

Hope it helps.
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I'm a little confused by this problem, or more so, the strategy for these type of coordinate geometry problems. I understand the problem conceptually and have read the GMAT CLUB book but for some reason, this problem was difficult.

For these type of problems, is it best just to plug in the points into the equation. Meaning, if I plugged in, I could see that 4, -8 could never be OK since y cannot be a negative number. That would be my initial reaction and therefore I would pick A as sufficient because the answer is a DEFINITE "no".

On the flip side, from looking at your graph, I can see how A is insufficient.

Sorry for the rather generic question but for some reason, the implementation for these types of problems is proving to be a little difficult for me.

Thanks!

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

In the xy-plane, does the line L intersect the graph of y = x^2

(1) Line L passes through (4, -8)
(2) Line L passes through (-4, 16)

There are 2 variables (slope, y-intercept) and 2 equations are given by the 2 conditions, so there is high chance (C) will be the answer.
C actually is the answer, but is too trivial.
If we look at the conditions separately,
condition 1 cannot determine anything
Condition 2 y=x^2 passes through (-4,16) so it's a 'yes' and is sufficient, making the answer (B).

For cases where we need 2 more equations, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.
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Given equation is that of a parabola. Y=X^2. In order for our statement to be true the points on line L, if they were to insect Parabola are to have the similar relationship between X and Y coordinates.

A is insufficient

B sufficient

Hence, B
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I guess there are no hard questions for Bunuel
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Bunuel if line L would intersect y=x^2 then coordinates of Line L should satisfy the equation y=x^2. But St 1 says the coordinates of L are (4,-8) 4^2=16 which doesn't equal -8. So how do the lines intersect then? there is a definite answer no. so A should also be satisfying?

Also, In DS questions info given in two statements arent contradictory. So coordinates of line L given in st1 arent satisfying the equation but coordinates of the same line in st 2 are satisfying. The line we are talking about is the same. How is this possible? (im sorry again, if my doubt is silly)

Edit: Understood how st 1 is insufficient. The line may or may NOT intersect.

Best here is to draw it out.

In the xy-plane, does the line L intersect the graph of y = x^2

(1) Line L passes through (4, -8)
Maybe, maybe not. x^2 is a parabola that extends ad infinitum for all points x ≥ 0 (i.e. the domain is negative to positive infinity)
Insufficient.

(2) Line L passes through (-4, 16)

y = x^2
16 = (-4)^2 = 16

Sufficient.

B.
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VeritasKarishma Just wanted to clarify a concept am having some doubts about pertaining to this question. Statement two clearly gives a yes as it satisfies y=x^2. But as for statement 1 it doesn't satisfy it i.e. it doesn't lie on the graph y=x^2 but we can't say its a clear NO(i.e Sufficient) because if two lines intersect they need to have only 1 solution in common i.e the line MAY still intersect the parabola.

What am trying to clarify is that for two lines to intersect we just need a common solution and not for all points (in this case 4,-8) to satisfy both equations?

Hope am able to explain my query. Looking forward to hear from you.
Best